Universit`
a degli Studi di Padova
Dipartimento di Fisica e Astronomia
Corso di Laurea Magistrale in Astronomia
A data-driven model for the cosmic merger
rate density of compact binaries
Master Degree Thesis
Candidate:
Filippo
Santoliquido
Supervisor:
Prof.ssa Michela Mapelli
Co-supervisor:
Dott. Nicola Giacobbo
The art of research is the sensitivity to decide when a useful and necessary simplification has become an obfus-cating oversimplification.
Abstract
Contents
1 Introduction 2
2 Aims 4
3 Astrophysics of compact objects 6
3.1 Single massive stellar evolution and SNe . . . 6
3.2 SN kicks . . . 8
3.3 Initial conditions . . . 9
3.4 Binary evolution . . . 10
4 Data-driven cosmological evolution 14 4.1 Cosmic star formation density . . . 14
4.2 Metallicity cosmic evolution models . . . 16
4.2.1 Rafelski et al., 2012 . . . 16
4.2.2 De Cia et al., 2018 . . . 18
5 The data-driven model 21 5.1 Merging efficiency . . . 21
5.2 MRD across cosmic time . . . 23
6 Results 28 6.1 DNS MRD Evolution . . . 29
6.2 BBH MRD Evolution . . . 30
6.3 BHNS MRD Evolution . . . 32
6.4 Comparisons with literature . . . 32
6.4.1 Neijssel et al. (2019) . . . 32
6.4.2 Mapelli & Giacobbo (2018) . . . 35
7 Conclusions 38 7.1 Further developments . . . 39
Appendices 41
A Adopted cosmology 41
Chapter 1
Introduction
The first direct gravitational wave (GW) detection - GW150914 - revolutionised the theoretical understandings of compact binaries. Astrophysicists learnt about the existence of binary balck holes (BBHs) that inspiral and merge within an Hubble time (Abbott et al., 2016). The inferred masses demonstrate that relatively heavy (MBH > 25M ) stellar black hole (BH) can form in nature. Since 2015 until 2017, during the first and the second observing run, the LIGO-Virgo collaboration (LVC) searched for new events and now the astrophysics community can refer to a catalogue that contains 11 merging compact binaries (The LIGO Scientific Collaboration et al.,
2018). The inferred parameters that describe this population are listed in table 1.1. This set of ten BBHs and one double neutron star (DNS) posed new questions regarding the formation channels of these detected binaries.
These objects possess a wide range of physical properties. For example, the light-est BBH so far detected is GW170608 with an inferred final mass of 17.8+3.2−0.7 M ; while GW170729 is the heaviest, having a final mass equal to 80.3+14.6−10.2 M . Fur-thermore, it is the most distant, since it merged at redshift z = 0.48+0.19−0.20. In April 2019, LVC began the third observing run (03), which will last one year. Therefore, several other GW detections are expected to enrich the above-mentioned catalogue. Each observing run has experienced improvements on sensitivity and these lead to an enhancement of the observing horizon. For instance, Advanced LIGO at design sensitivity can observe two merging black holes with masses (30 + 30)M out to a cosmological redshift equal to z ∼ 1 (Abbott et al.,2018). The observatory network will search for mergers in the entire Universe. Hence, it is essential to predict the double compact object (DCO) merger rate density (MRD) across cosmic time.
I developed a model that evaluates the MRD from merging DCO catalogues obtained with population-synthesis simulations. Isolated binary evolution was per-formed using MOBSE, which is an updated version of the widely used binary pop-ulation synthesis code BSE (Giacobbo et al., 2018b). In chapter 3, I will carefully describe the input parameters, the involved physical processes and the results of the population-synthesis simulations. I placed merging DCO obtained from isolated binary evolution in a cosmological context. The star formation rate reported in
Event m1 M m1 M Mchirp M χef f Mf M dL Mpc z ∆Ω deg2 GW150914 35.6+4.8 −3.0 30.6+3.0−4.4 28.6+1.6−1.5 −0.01+0.12−0.13 63.1+3.3−3.0 430+150−170 0.09+0.03−0.03 180 GW151012 23.3+14.0 −5.5 13.6+4.1−4.8 15.2+2.0−1.1 0.04+0.28−0.19 35.7+9.9−3.8 1060+540−480 0.21+0.09−0.09 1555 GW151226 13.7+8.8−3.2 7.7+2.2−2.6 8.9+0.3−0.3 0.18+0.20−0.12 20.5+6.4−1.5 440+180−190 0.090.04 −0.04 1033 GW170104 31.0+7.2−5.6 20.1+4.9−4.5 21.5+2.1−1.7 −0.04+0.17−0.20 49.1+5.2−3.9 960+430−410 0.19+0.07−0.08 924 GW170608 10.9+5.3−1.7 7.6+1.3−2.1 7.9+0.2−0.2 0.03+0.19−0.07 17.8+3.2−0.7 320+120−110 0.07+0.02−0.02 396 GW170729 50.6+16.6−10.2 34.3+9.1−10.1 35.7+6.5−4.7 0.36+0.21−0.25 80.3+14.6−10.2 2750+1350−1320 0.48+0.19−0.20 1033 GW170809 35.2+8.3−6.0 23.8+5.2−5.1 25.0+2.1−1.6 0.07+0.16−0.16 56.4+5.2−3.7 990+320−380 0.20+0.05−0.07 340 GW170814 30.7+5.7−3.0 25.3+2.9−4.1 24.2+1.4−1.1 0.07+0.12−0.11 53.4+3.2−2.4 580+160−210 0.12+0.03−0.04 87 GW170817 1.46+0.12 −0.10 1.27+0.09−0.09 1.186+0.001−0.001 0.00−0.01+0.02 ≤ 2.8 40+10−10 0.01+0.00−0.00 16 GW170818 35.5+7.5 −4.7 26.8+4.3−5.2 26.7+2.1−1.7 −0.09+0.18−0.21 59.8+4.8−3.8 1020+430−360 0.20+0.07−0.07 39 GW170823 39.6+10.0 −6.6 29.4+6.3−7.1 29.3+4.1−3.2 0.08+0.20−0.22 65.6+9.4−6.6 1850+840−840 0.34+0.13−0.14 1651
Table 1.1: The eleven confident detections provided by The LIGO Scientific
Col-laboration et al.(2018). Relevant parameters and median values with 90% credible intervals are reported. For binary black hole (BBH) events the redshift was calcu-lated from the luminosity distance and assumed cosmology as discussed in Appendix A. The columns show source frame component masses mi, chirp mass Mchirp, di-mensionless effective aligned spin χef f, final source frame mass Mf in, luminosity distance dL, redshift z and sky localisation ∆Ω. The sky localisation is the area of the 90% credible region.
2012;De Cia et al.,2018) reproduce a data-driven cosmological evolution.
The distribution of binary coalescences as a function of redshift has been inves-tigated by several authors. In particular, I refer to Dominik et al. (2013). They expressed the metallicity evolution with redshift using a mass-metallicity relation, where the distribution of galaxy masses was given by a Schechter-type probability density function. This last choice is redshift-limited to z ∼ 4, while metallicity evolution model based on damped Ly-α systems (DLAs) can reach up to redshift z ∼ 5 (Rafelski et al. (2012), hereafter R2012). Baibhav et al. (2019) investigated the MRD across cosmic time using this approach. I extended my results considering another metallicity evolution model (De Cia et al.(2018), hereafter D2018) which is consistent with the measured solar metallicity in the local universe (Gallazzi et al.,
Chapter 2
Aims
The MRD as a function of cosmological redshift is an essential quantity in GW astronomy. In these years, it is gaining increasing importance because the LVC has planned to improve the sensitivity of GW observatories and enlarge the observatory network.
Among other improvements, I would like to report two facilities which pertain the next future. The European gravitational observatory (EGO) is planning to build the Einstein Telescope (ET), while the US contribution is represented by the Cosmic Explorer (CE). These two new ground-based observatories belong to third generation (3G) observatory network (Reitze et al., 2019). The main characteristic of these new GW interferometers is their capability to observe a merging BBH up to redshift z ∼ 10. This means that these new instruments can provide data from the early Universe, since the first stars are known to form at redshift z ∼ 7 (Sobral et al.,
2015). Hence, the whole evolutionary life of stellar-born DCOs can be measured. The Einstein Telescope and the Cosmic Explorer will start operating from 2030. A physical quantity that will be fully provided by measures will be the MRD across cosmic time. Until that date, theoretical astrophysicist efforts must aim to assess the predictive power of their models.
In order to evaluate the MRD across cosmic time, the population-synthesis sim-ulations must be put in a cosmological context. In other words, I must convolve the results of population-synthesis simulations as delay time distributions, with star for-mation rate (SFR) and metallicity evolution model. The delay time is the amount of time elapsed from the formation of a compact binary until its coalescence. The delay time distribution depends on physical processes that determine the binary evolution. In chapter 6, I will show how common envelope (CE) ejection efficiency and the natal kick velocity distribution change the value of the MRD. It is now evident how the MRD across cosmic time is essential to track the formation and evolution of a compact binary.
Chapter 3
Astrophysics of compact objects
This Thesis can be regarded, from the point of view of the methodology, as a natural prosecution of Giacobbo and Mapelli (2018). In this paper, the authors evaluated the MRD in the local universe (z = 0.1) and compared it with the confident interval provided by LVC. In order to obtain the MRD across the entire history of the Uni-verse, I appropriately modified the equation that combines merging DCO catalogues with cosmological evolution. These modifications are shown in chapter 5.
The above-mentioned paper and this thesis share the population-synthesis simu-lation. In this chapter, I describe the underlying physics that drives a binary system merger; starting from single massive star evolution and supernova (SN) engines, to SN kicks and binary evolution. This chapter wants to provide an outline of the astrophysics of double compact objects.
3.1
Single massive stellar evolution and SNe
Hot massive stars are characterised by strong stellar winds. MOBSE adopted a description of the mass loss ( ˙M = dM
dt ) as ˙ M ∝ Zβ
where Z is stellar metallicity and β varies depending on the Eddington factor (Γe), as follows β = 0.85, if Γe< 2/3 β = 2.45 − 2.4 · Γe if 2/3 ≤ Γe< 1 β = 0.05 if Γe≥ 1
This reflects that mass loss becomes almost insensitive to metallicity, in case a massive star approaches or exceeds its Eddington limit, which corresponds the star to be radiation pressure dominated. Figure 3.1 shows the relation between the maximum mass of the compact remnant and the metallicity.
Figure 3.1: Mass of the compact object (Mrem) as a function of the mass of the progenitor star (MZAMS) for 8 metallicities between Z = 0.02 and 0.0002. See
Giacobbo et al. (2018a) for details. The curves have been obtained using MOBSE. The rapid SN explosion model is included, as well as the PISNe prescription. Indeed, remnant masses do not exceed the mass gap lower limit (∼ 60 M )
(Littenberg et al., 2015). The upper mass gap is between 60 M and 120 M , and it is a consequence of pair-instability supernovae (PISNe) (Spera and Mapelli,
2017). Figure 3.1 shows that no stars leave remnants with Mrem > 60 M , which corresponds to the mass gap lower limit.
The lower mass gap, between neutron star (MN S < 2 M ) and black holes (MBH > 5 M ), can be be reproduced by adopting the prescription of the rapid core-collapse supernovae (CCSNe) mechanism (Belczynski et al., 2012). Giacobbo and Mapelli(2018) considers only the rapid SN model for each simulation, as shown in table 3.1.
Neutron stars (NSs) are also expected to form from via electron-capture super-novae (ECSNe). Indeed, if a growing degenerate oxygeon-neon (ONe) core reaches the mass of
MECSN = 1.38M
3.2
SN kicks
When a star explodes as a SN and then leaves a remnant, the latter receives a ve-locity kicks. Any asymmetries in the explosion mechanism can generate this kick. Evidences for such phenomenon come from observations of radio pulsars with exces-sive velocity with respect to field stars (Hobbs et al.,2005). The velocity distribution is better represented by a Maxwellian function, defined as follows
f (v, σ) = r 2 π v2 σ3 exp − v 2 2σ2 (3.1) where v ∈ [0, ∞). Natal kicks are one of the most debated issues about compact objects, since they have crucial effect on the evolution of DCO. The kicks can either disrupt the binary system or change its orbital properties. A parameter that is affected by these kicks is the eccentricity. In section 3.4, I explain the impact of an enhanced eccentricity on merging efficiency.
Natal kick average velocities are believed to depend on SN mechanisms. Indeed, ECSNe is a rapid process with respect to CCSNe, and thus the asymmetries that generate the natal kick have less time to develop the kick. The consequence is that ECSNe-born NSs are expected to have low natal kicks. Therefore, when an ECSNe occurs, MOBSE draws the SN kick velocity from a Maxwellian distribution (see equation 3.1) defined by a root-mean square equal to
σECSN = 15 km s−1
corresponding to an average velocity hviECSN ∼ 23 km s−1. In contrast, NSs born from CCSNe are believed to receive higher kicks. Hence, MOBSE randomly draws the kicks from a Maxwellian velocity distribution characterised, in this case, by a root-mean square equal to
σCCSN = 265 kms−1
which corresponds to an average velocity hviCCSN ∼ 420 km s−1. However, several studies claim that this velocity distribution underestimates the number of NSs that receive low natal kicks despite the CCSNe mechanism that generates them. For instance, ultra stripped SNe eject a little amount of mass (< 0.1M ) and thus the kick is also small (< 50 km s−1).
Given these uncertainties, Giacobbo and Mapelli (2018) decided to simulate two extreme cases for CCSNe kicks. The high-velocity kicks are drawn from a Maxwellian distribution defined by σCCSN = 265 kms−1; and the low-velocity kicks are defined by the Maxwellian distribution that has the same root-mean square of ECSNe kicks. Table 3.1 lists the abbreviations assumed henceforth for the various parameter choices.
Natal kicks have higher impact on NS mergers, because they receive the full kick. In contrast, BHs receive a kick velocity (vBH) that is reduced by the amount of the fallback as
vBH = (1 − ff b)v
Abbreviation α σECSN σCCSN SN km/s km/s α1 1.0 15.0 265.0 Rapid α3 3.0 15.0 265.0 Rapid α5 5.0 15.0 265.0 Rapid CC15α1 1.0 15.0 15.0 Rapid CC15α3 3.0 15.0 15.0 Rapid CC15α5 5.0 15.0 15.0 Rapid
Table 3.1: Different parameter prescriptions of the adopted MOBSE simulations
(Giacobbo and Mapelli,2018). The aim was to test the effect of different natal kick velocity distributions and CE ejection efficiencies on compact binaries evolution.
stellar envelope that falls back to the remnant. For the most massive BHs, which are formed silently, without a SN explosion and hence via direct collapse, the fallback parameter is equal to ff b= 1 (Fryer et al., 2012).
3.3
Initial conditions
A binary system is defined by the masses and metallicity of the two stars, the period (or semi-major axis) and the eccentricity of the orbit. The mass of the primary star (m1) is randomly drawn in MOBSE from the Kroupa initial mass functions (Kroupa,
2001), that is
z(m1) ∝ m 2/3 1
The allowed mass range is [5M − 150M ]. For this set of simulations, it was convenient that MOBSE simulated only massive star. The upper mass cutoff is determined by technical issues. Other population-synthesis code can overcome this limit; for example SEVN (Spera et al.,2015). The mass ratio between the secondary and the primary member of the binary (q = m1m2) follows the distribution found by
Sana et al.(2012) and it is equal to
z(q) ∝ q−0.1
with q ∈ [0.1 − 1]. The orbital period P and the eccentricity e distributions are also extracted according to Sana et al.(2012), they are equal to
z(P ) ∝ P−0.55 where P = log10(dayP ) ∈ [0.15 − 5.5]; and
z(e) ∝ e−0.42 with e ∈ [0, 1].
Z 0.0002 0.0004 0.0008 0.0012 0.0016 0.002 0.004 0.006 0.008 0.012 0.016 0.02
Table 3.2: Each binary system is considered to evolve with the same initial metal
content. These are the 12 metallicity values with which the adopted MOBSE simu-lation was run
3.4
Binary evolution
MOBSE adopts a description of binary evolution similar toHurley et al.(2002). The evolution of two stars in a binary system is characterised by particular features. The physical processes that mainly affect the binary evolution are the mass transfers, the CE phase and the GW emission.
In a binary system, a star can accrete material from the companion in case the latter manifests mass loss due to stellar wind. The average accretion rate on the secondary star, can be estimated according to Hurley et al.(2002) as
h ˙m2Ai = −√ 1 1 − e2 Gm2 v2 w αw 2a2 1 (1 + v2)32 ˙ m1W
where h ˙m2Ai is mass accretion rate on the secondary star, e the eccentricity, vw is the wind velocity, αW is an efficiency constant, a the semi-major axis, ˙m1W is the mass loss rate from the primary star and v2 is equal to
v2 = v 2 orb v2
W
where vorb is the orbital velocity. Since ˙m1W is quite low (< 10−3M yr−1) and vw is usually high (> 1000 kms−1) with respect to the orbital velocity, this kind of mass transfer is usually rather inefficient (Mapelli,2018). It is although essential to correctly predict the orbital evolution of the binary. Indeed, most of the mass lost by a companion usually does not accrete onto the other, and that translates in an angular momentum loss, which implies circularisation of the orbit.
In contrast, Roche lobe overflow (RLOF) is a mass transfer mechanism that can considerably affect the binary evolution. The Roche lobe is the equipotential surface that surrounds a star which belongs to a binary system. Eggleton (1983) provided the following approximation for the Roche Lobe of the primary star
rL,1= a
0.49q2/3
0.6q2/3+ log(1 + q1/3)
Figure 3.2 shows three different configurations that consider an increasing grad-ual filling of the Roche lobes (Schmidt,2016). Figure 3.2(a) shows a detached binary, i.e. each star of the binary system evolves independently unless wind mass transfer sets in. When the most massive star, the primary star, overfills its own Roche lobe, a stable mass transfer to secondary star takes place. This configuration is called semi-detached and it is represented by figure 3.2(b). When an unstable mass trans-fer sets in or both stars overfill their Roche lobe, a contact binary is formed, as is shown in figure 3.2(c). The subsequent evolution of this type of binary is driven by CE phase (Ivanova et al.,2013). However, if the donor that enters the CE phase is a main sequence star or a Hertzsprung-gap (HG) star, the binary system is forced to merge in MOBSE. Indeed, these type of stars has not yet developed a steep density gradient between the core and the envelope.
The CE phase is described by the αλ formalism, which is the most common formalism adopted to model CE. The basic idea is that the energy needed to eject the envelope comes uniquely from the loss of orbital energy of the two cores during the spiral in due to frictional forces, that is
α∆Eorb = Ebind
λ (3.2)
where α quantifies the available energy to unbind the envelope, while λ measures the mass distribution of the envelope. The former is a free parameter, while the letter depends on stellar type (i.e. mass and luminosity), also to account for the contribution of recombinations (Claeys et al.,2014), which are another energy source to unbind the envelope. Therefore, the fate of a binary that entered the CE phase is determined by the following equation
α Gm1m2 2ai + Gm1,cm2,c 2af = G λ m1m1,env R1 + m2m2,env R2 (3.3) where ai and af are the initial and final binary separation, m1 and m2 are the initial masses of the primary and secondary star respectively, m1,c (m2,c) is the primary (secondary) star remnant mass that lost its envelope with mass equal to m1,env (m2,env). R1 and R2 are the primary and secondary star radius, respectively.
Higher is the α parameter, more efficient is the energy transfer to the envelope, that is the binding energy loss mechanism is more efficient. This implies a less inspiral of the binaries when α > 1, that is the probability that the binary survives the CE phase is higher. In order to test the effect of CE on the merging efficiency of DCOs, the adopted population-synthesis simulations assume three different values of CE efficiency, namely α = 1, 3 and 5.
Table 3.1 recapitulates the different simulations considered in this Thesis, with the abbreviation used in following legends.
(a) Detached binaries
(b) Semi-detached binaries
(c) Contact binaries
Peters(1964) showed that a binary system is induced to merge because of energy loss from GW emission. The following equation expresses the semi-major axis (a) diminution as a function of binary system parameters, due to GW emission:
da dt = − 64 5 G3m 1m2(m1+ m2) c5(1 − e2)72 a−3 (3.4)
Chapter 4
Data-driven cosmological evolution
A satisfactory and complete evolution model of the Universe requires several physical concepts as well as high performance computer facilities. Numerical cosmological simulations are known to be one of the most difficult challenges that astrophysicists have to face. Hydrodynamics is the most representative of these challenges.
In order to put population-synthesis simulations in a cosmological context as quickly as possible, the cosmological evolution can be simplified. Indeed the latter can be represented only by the SFRD and a metallicity evolution model. Since DCOs are stellar born, their number is expected to depend on star formation rate. Similarly, I illustrated in the previous chapter how metallicity can affect the remnant mass and, thus, DCO merger efficiency.
Therefore, I selected the two main contributors to DCO cosmic evolution. In order to better constrain the cosmic evolution, I chose analytic functions directly inferred by data. In this chapter, I will schematically describe these functions, which are the Madau and Dickinson (2014) star formation rate density which represents the state-of-the-art of the subject; and two metallicity evolution models, namely
Rafelski et al.(2012) and De Cia et al. (2018).
4.1
Cosmic star formation density
Conceptually, measuring the star formation rate means inferring mass content from integrated light from galaxies. In order to achieve this,Madau and Dickinson(2014) decided to define a conversion factor Kν that relates the specific luminosity Lν to the SFR. These quantities are operationally related by a convolution
SFR = Kν × Lν
(a) FUV (top) and IR (bottom) data
(b) FUV+IR
Figure 4.1: The history of cosmic SFR from Far-UV and IR separately (a) and
The most common tracers of SFR are the UV emissions from massive and young stars and IR radiation. The latter comes from dust emission. These IR photons are those absorbed as UV photons and remitted by dust. Spectra also contains SFR indicators. They are Hα and Lyα. These lines come from recombination due to ionised gas by UV photons.
Madau and Dickinson(2014) considered only surveys that measured SFRs from far-UV (FUV, generally 1500 ˚A) or mid-IR (MIR) and far-IR (FIR). For their anal-ysis, they stopped at redshift z ∼ 8. Figure 4.1 shows the cosmic SFRD (Ψ(z)) from UV and IR data. The best-fitting function is given by the following equation:
Ψ(z) = 0.015 (1 + z) 2.7 1 +h(1+z)2.9 i5.6 [M year −1 Mpc−3] (4.1)
Coming from the past, a rising phase is noticeable. From redshift z ∼ 8 to redshift z ∼ 3, the SFRD scales as Ψ ∝ (1 + z)−2.9. It slows down and peaks at some point between z = 2 and 1.5, when the Universe was ∼ 3.5 Gyr old. This is followed by a gradual decline to the present day, roughly as Ψ ∝ (1 + z)2.7. The comoving SFR density at redshift z ∼ 7 was approximately the same as that measured today.
The data-driven cosmological simulation considered in this Thesis adopts Equa-tion 4.1 as expression of the cosmic SFRD.
4.2
Metallicity cosmic evolution models
In this section, I will briefly describe the two adopted metallicity evolution models. These two works required mainly observational efforts. Indeed, they complete the data-driven contributions to my model. I decided to avoid explaining observational details and data reduction procedure, although essential for the results, for the sake of brevity.
4.2.1
Rafelski et al., 2012
The main purpose ofRafelski et al.(2012) is to trace the build-up of heavy elements in neutral gas across cosmic time, by investigating the metal abundances of DLAs, out to z ∼ 5. This particular class of absorption quasars was chosen because it contains mainly neutral gas. Therefore, DLAs dominate the neutral gas content of the Universe, and exhibit properties indicating that they are neutral-gas reservoirs for star formation at high redshift.
Rafelski et al.(2012) measured the metal abundances of 47 DLAs from z ∼ 0.09 to z ∼ 5.06; it is remarkable that 30 of these systems are at redshift z > 4. Their data were collected with two instruments on the Keck I and II 10-meter telescope, namely the Echellette Spectrograph and Imager (ESI) and the High Resolution Echelle Spectrometer (HIRES). They also combine their metallicity measurements with 195 DLAs from previous surveys, in order to enlarge the sample.
Among other features, this plot shows an evident dispersion at all redshifts. The dispersion in [M/H] (1) is ∼ 0.5 dex and does not appear to evolve with redshift.
Rafelski et al.(2012) stated that this is not an observational error, but is due to an intrinsic scatter amongst the DLAs. This result was confirmed by successive works, as I will show in the next section (4.2.2).
Another feature of figure 4.2 is that the average [M/H] decreases with increasing redshift. The evolution of [M/H] was investigated by computing the cosmological mean metallicity hZi, in the following way
hZi = log P i10 [M/H]iN (HI) i P iN (HI)i (4.2) where i represents each bin of DLAs as a function of redshift. Each bin contains an equal number of DLAs, that is 26 DLAs/bin. NHI is the HI column density. Rafelski
et al.(2012) confirmed that each DLA has NHI > 2 × 1020cm−2 based on the SDSS spectra with spectral resolution FWHM ∼ 2 ˚A.
The black dotted line in figure 4.2 is a linear fit to the hZi values. After per-forming a χ2 minimisation, they found that
hZi = (−0.22 ± 0.03)z − (0.65 ± 0.09) (4.3) This result is in accordance with previous studies, as pointed out byRafelski et al.
(2012).
They performed an analysis that completely neglects dust obscuration. Dust depletion contribution on metal abundances was assessed byDe Cia et al. (2018). I present their paper in the next section (4.2.2).
4.2.2
De Cia et al., 2018
De Cia et al.(2018) also provided metal abundances of DLAs from absorption line spectroscopy. They did not neglect the presence of dust, which can affect the metal abundance measurement up to 0.5 dex. The improvement they achieved in this paper was to provide a simplified way, called single-reference method, for calculating dust correction.
In a previous paper (De Cia et al.,2016), they found dust-corrected metallicities based on the simultaneous study of the abundances of several metals. This is the most solid method. In contrast, the single-reference method relies only on a single abundance relative to Iron (Fe), that is [X/Fe] . In this case, the dust-corrected metallicity is defined as follows
[Fe/H]tot = [Fe/H] − δF e (4.4)
(1)Rafelski et al.(2012) adopted the standard notation where the metal abundance [M/H] signifies
Figure 4.3: Dust-corrected metallicities [M/H]tot derived from several metals si-multaneously in De Cia et al. (2016) (black filled circles) and the dust-corrected metallicities [Fe/H]tot calculated with the single-reference method, as shown in equa-tion 4.4, for the large sample (gold open symbols). The shape of the symbols shows which reference was used for dust correction, as labelled in the legend. The black solid and dotted line display the linear fit to the data and the intrinsic scatter of the relation, respectively. The large open purple circles show the mean DLA metal-licities weighted for the N (HI) content, in bins of redshift and the linear fit to those points is shown by the dotted-dashed purple line (expressed by equation 4.5). The dashed blue curve shows the average DLA metallicity derived by Rafelski et al.
where δF e is the dust depletion of Fe, computed as δFe = A2Fe+ B2Fe× [X/Fe]
where X is the element on which dust correction is performed, which it can be either Zinc (Zn), Silicon (Si) or Sulphur (S). The coefficients are equal to A2Fe = −0.01 ± 0.03 and B2Fe = −1.26 ± 0.04. They applied this single-reference method dust corrections on a large sample, that contains 236 DLAs. The size of this sample is comparable to the one on whichRafelski et al. (2012) performed their analysis.
In Figure 4.3, De Cia et al. (2018) plotted the solid metallicity measurements (black dots) from De Cia et al. (2016), for comparison with the ones obtained with the single-reference method, illustrated in the figure with open gold symbols. They fit a linear regression to these data, although they are not weighted for N (HI) content. They must be weighted to avoid giving too much importance to the low-metallicity systems which carry less gas. Therefore, they computed the weighted neutral gas cosmic average metallicities as in equation 4.3. To this purpose, they divided the sample in bins of redshift (z < 1, 1 ≤ z ≤ 2, 2 ≤ z ≤ 3, z ≥ 4). The weighted metallicity values are shown in figure 4.3 as purple circles. They performed a linear fit on these data, which yields
hZi = (−0.24 ± 0.14)z + (−0.18 ± 0.21) (4.5) A consistent comparison can be now done with the metallicity evolution model provided by Rafelski et al. (2012) (see equation 4.2 and dashed blue line in figure 4.3). The two slopes of the metallicity evolution models are very similar, but dust correction provided an intercept 0.4-0.5 dex higher. It approaches solar metallicity (Z = 0.02) at redshift zero. This result is rather important, since Gallazzi et al.
(2007) estimated from the Sloan Digital Sky Survey (SDSS) that the average metal content (Z∗) in the local Universe (hz ∼ 0.1i) is consistent with solar metallicity, that is
hZ∗i = 1.04 ± 0.14 Z where Z is assumed equal to 0.02.
This relation of metallicity with redshift also shows a large scatter (∼ 0.5 dex). As I mentioned in the previous section,Rafelski et al. (2014) found a similar result. At any given redshift, we expect that DLAs may select galaxies with a range of different masses and metallicities. The scatter of metallicity versus redshift is there-fore physical and it reflects a spread in metallicity. Considering a mass metallicity relation (Tremonti et al., 2004), a galaxy with little mass will contribute less to the average metallicity evolution.
Chapter 5
The data-driven model
In this chapter, I will describe the procedure to calculate the MRD across cosmic time. Appendix B includes the Python script that implements this methodology.
5.1
Merging efficiency
For each simulation set (see table 3.1), I evaluated the number of merging DCOs per unit mass, also known as merging efficiency and denoted as η, in the following way
ηi = fbinfIMF
Nmerger,i Mtot,sim
[M−1 ] (5.1)
where i stands for the three different classes of DCOs, which are DNS, BBH, BHNS. Nmerger,i is the number of DCOs that merge within an Hubble time. The latter is equal to
tH0 = 1 H0
= 14.4 ± 0.2 Gyr
where the Hubble constant H0 = 67.8 ± 0.9 km s−1Mpc−1. Mtot,sim is the total initial mass of the simulated stellar population. This quantity is different for every simulation set. Indeed, the number of simulated binaries is fixed and corresponds to 1.2 × 108 binary systems, but the total initial mass is randomly drawn from the IMF, as explained in section 3.3. In addition, fIMF = 0.285 is a correction factor that takes into account the fact that the MOBSE simulation adopted in this thesis, considered only systems which have primary component more massive than 5 M . Another correction factor is fbinwhich takes into account the fact that the simulation evolves only binary systems. Assuming that 50 % of the stars are in binary systems, fbin = 0.5 (Sana et al., 2012)). Figure 5.1 shows η as a function of the metallicity Z for each simulation set and separately for each class of DCOs (Giacobbo and Mapelli, 2018). The number of mergers per unit mass spans a large range of values depending on natal kicks, on CE efficiency and progenitor metallicity.
Figure 5.1: Number of merging DCOs per unit mass (η) as a function of pro-genitors metallicity for all the adopted simulations (see table 3.1). Top-left panel: merging DNSs; bottom-left panel: merging BHNS; bottom-right panel: merging BHBs (Giacobbo and Mapelli,2018).
at high metallicity. Similarly, ηBHN S & 2 orders of magnitude greater at Z = 0.0002 rather than at Z = 0.02. This is due to the strong dependence of equation 3.4 on remnant masses (∝ m3), which are in turn strongly affected by progenitor metallicity, as shown in figure 3.1. Hence, binary systems born in a metal-poor environments tend to merge more efficiently.
5.2
MRD across cosmic time
The delay time, that is the time elapsed between a formation of a binary and its coalescence is defined in MOBSE as follows
tdelay = tevol+ tGW (5.2)
where tevol is the time elapsed from the binary system formation as two ZAMS stars, until the collapse of both progenitors. In the isolated evolution environment, if the binary system survives the CE phase, the only mechanism that can disperse total energy is GW emission. This energy loss mechanism is described by equation 3.4. From this equation, it is possible to derive the expression for the time elapsed for disperse the total energy by GW emission, which yields
tGW = 5 256 c5 G3 a4(1 − e2)7/2 m1m2(m1+ m2) (5.3) This amount of time is exceedingly higher than the time the massive binary system takes to evolve. For example, in a typical situation, a BBH with masses (10−10) M , with initial major-axis a = 10 R and eccentricity e = 0.5, takes tGW ∼ 3 Gyr to merge.
Figure 5.2 shows the delay time distribution (Tdelay) for each simulation set. The contribution of initial metallicities are summed over in these plots. It is worth noting that number of merging BBHs is always greater than the number of the other merging DCOs. This is due the fact that tGW (see equation 5.3) strongly depends on remnant masses, namely tGW ∝ m3. In other words, since BBHs are always more massive than the other DCO types, this translates in an higher merging efficiency. The Tdelay trend is slightly affected by the three α parameters and the natal kick velocity distributions. Indeed, in any case it is compatible with dN/dt ∝ t−1. This function is represented by the dot-dashed black line in figure 5.2. The binary evolution free parameters affect mainly the total number of merging DNSs, as shown previously in figure 5.1. For instance, CC15α5 that is an efficient CE ejection and a low natal kick velocity distribution, provides a DNS delay time distribution that is comparable with the BBH one.
The BBH delay time distribution is mainly affected by the metal content of the progenitor stars. Figure 5.3 shows the delay time distribution obtained with an efficient CE prescription and fast natal kicks (α5, 5.3(a)) and slow natal kicks (CC15α5 ,5.3(b)). Different initial metallicities (table 3.2) are grouped in two sets: low metallicities (Z < 0.0012) and high metallicities (Z > 0.0016). It is worth noting that the majority of BBHs that are born from metal-poor stars tend to merge with short delay time (td < 1 Gyr). BBHs that merge with long delay time (td> 1 Gyr) are born from stars with initial metallicity greater that Z > 0.0016. This trend is even more evident for systems with considerably high delay time (tdelay > 6 Gyr). Indeed, merging system fraction remains constant for metal-poor progenitors, while drops to zero for metal-rich progenitors. Lower SN kick velocities mildly affect the delay time distribution slopes. Indeed, BHs receive natal kicks that are reduced by the fallback parameter, as shown in section 3.2.
0 2 4 6 8 10 12 14 Delay time [Gyr]
10 2 10 1 Co un ts/ NTOT Z<0.0012 Z>0.0016
(a) BBHs - α5 - high kicks
0 2 4 6 8 10 12 14
Delay time [Gyr] 10 2 10 1 Co un ts/ NTOT Z<0.0012 Z>0.0016 (b) BBHs - CC15α5 - low kicks
Figure 5.3: Delay time distributions for BBHs obtained with an efficient CE
Paper Abbreviation Metallicity evolution model
Rafelski et al.(2012) R2012 hZi = (−0.22 ± 0.03)z − (0.65 ± 0.09)
De Cia et al. (2018) D2018 hZi = (−0.24 ± 0.14)z − (0.18 ± 0.21)
Table 5.1: Metallicity evolution models as a function of redshift considered for the evaluation of the MRD across cosmic time. The first column shows the reference paper, the second the abbreviation adopted in the forthcoming legends, and the third the results of linear fits on observational data. I implemented the conversion Z = 0.02 · 10hZi to match the scale of the initial metallicities reported in table 3.2.
high redshift is associated with initial low metallicity of the progenitors. DCOs that form in different metallicity environments are expected to have different delay time distributions as well as different formation redshifts.
Table 5.1 recapitulates the two metallicity evolution models adopted in the forth-coming analysis. They express the average metal content of the Universe as linear functions of cosmological redshift.
The procedure, I developed to evaluate the cosmic MRD, consists on counting how many DCOs merge at a given redshift z that formed in every higher redshift (z0 > z). To this purpose, I defined a redshift binning that goes from z = 15 to z = 0 with a constant step equal to ∆z = 0.1. In order to compute the total mass of stars formed within a specific redshift bin, I considered the SFRD presented in section 4.1. This quantity must be multiplied by ηi in order to obtain the fraction of the total merging mass at a given redshift. I then defined the merger time tmerger as
tmerger(z0, z) = tf orm(z0) − tdelay(z, Z(z0)) (5.4) tdelay is the delay time of a single binary generated by the simulation associated with initial metal content Z, which is derived by eitherRafelski et al. (2012) or De Cia et al.(2018) metallicity models, computed on z0. Since the initial metallicity set (see table 3.2) is not linearly evenly spaced, I had to choose the delay time distribution associated with the closest metallicity available. This implies a non-smooth MRD evolution as a function of redshift (see chapter 6). tf orm is the formation time corresponding to the lookback time computed at formation redshift z0. The lookback time is given by the following equation:
tlb(z) = 1 H0 Z z 0 dz0 (1 + z0)pΩm(z + 1)3+ Ω k(1 + z)2+ ΩΛ (5.5)
where Ωm, Ωk and ΩΛ are the density parameters (see appendix A for the adopted cosmology andHogg (1999) for reference).
The merger time distribution Tmerger, computed with the values in equation 5.4, must be divided by Nmerger,i, the total number of merging objects, in order to obtain the fraction of merging DCO at given formation redshift bin. I implemented the concepts developed thus far in the following integral:
where dztlb0 is given by dtlb dz0 = 1 H0 1 (1 + z0)pΩ m(z0+ 1)3+ (1 − Ωm) (5.7)
and tlb(z) is expressed as in equation 5.5 with Ωk = 0 since a flat Universe is considered. I must multiply by the lookback time because the relation between redshift and time is not linear.
Chapter 6
Results
In this chapter, I will present the results obtained evaluating equation 5.6 for each population-synthesis prescription (see table 3.1), for each metallicity evolution model (see table 5.1) and for each class of DCOs, namely DNSs, BHNSs and BBHs. I will also provide a comparison of my results with those obtained by two previous works (see section 6.4).
Each figure presented hereafter includes the LIGO-Virgo confidence intervals, presented in the first gravitational-wave transient catalogue (GWTC-1) of compact binary mergers. The GW events were detected by LVC interferometers, during the first and the second observing run (The LIGO Scientific Collaboration et al., 2018). The merging rates inferred from GW detections at 90% credible intervals are shown in table 6.1. These merger rate intervals are limited to the observing horizon of LIGO and Virgo, which is currently z ∈ [0, 1]. The detectability of a merging binary depends naturally on a series of parameters, for example the merging masses. A BBHs of mass (10 − 10)M can be detected up to redshift z ∼ 0.2 (Baibhav et al.,
2019). Since there are too few GW detections so far in the redshift interval z ∈ [0, 1], LVC do not consider a MRD evolution with redshift. Therefore, the credible intervals shown in table 6.1 have to be considered redshift-independent.
Every figure presented in the following sections, from 6.1(a) to 6.3(b), has some common features.
DCO class MRD [Gpc−3y−1] DNS [110 - 3840] BBH [9.7 - 101]
BHNS [ - 610]
Table 6.1: Merger rates inferred from the LIGO-Virgo detections for the three
classes of DCOs, at 90% credible intervals (The LIGO Scientific Collaboration et al.,
Each cosmic MRD shows a similar trend, namely it increases at increasing red-shift up to a peak and then decreases. If metallicity did not play a role and if delay time was extremely short, we would expect a peak of MRD coincident with the maximum of the SFRD, that is at z ∼ 2 (see equation 4.1). When α5 CE ejec-tion efficiency is considered, each DCO class shows its MRD that peaks at z ∼ 2, see figure 6.1 for DNSs, figure 6.2 for BBHs and figure 6.3 for BHNS. This trend is independent on the assumed metallicity evolution model. However, some MRD peaks are anticipated or delayed with respect to the SFRD maximum. This can be explained as follows. First, some merging efficiencies strongly depend on progeni-tor’s metallicity, especially BBHs and BHNSs, but even some models of DNSs show a mild trend with Z, see figure 5.1. Second, the distribution of delay times depends on several population-synthesis features and on metallicity, as I showed in figure 5.2 and 5.3. For instance, CC15α5 in figure 6.1(b) shows a maximum at redshift z ∼ 1, while CC15α1 peaks at z ∼ 1.4. In contrast, BBHs tend to merge mainly at redshift z > 2. Figure 6.2(a) shows the peaks at redshift z > 2 for α1 and α3, independently on the natal kick velocity distributions. This trend is still more evident once De Cia et al. (2018) (hereafter D2018) metallicity evolution model is considered (see figure 6.2(b)). CC15α1 and α1 population-synthesis prescriptions are responsible for MRD peaks close to redshift z ∼ 3. This is due to the fact that D2018 model selects at the redshift where the SFRD peaks, binary population that have an higher initial metallicity, with respect toRafelski et al. (2012) metallicity evolution model (hereafter R2012). These binaries eventually merge with long delay times, and this explains why the SFRD peak is lowered with respect to R2012 model. Figure 5.3 shows that the fraction of merging BBHs that have delay times longer than 10 Gyr is still consistent, only if they are born from metal-rich progenitor stars .
Another common feature is the MRD non-smooth trend. A part from the MRD evaluated with α5 prescription for DNS, which is undoubtedly continuous, the other functions are segmented. This is due to the fact that the population-synthesis sim-ulation initial metallicities are not linearly spaced, and thus the metallicity changes abruptly at increasing redshift. This fact clearly is not physical and it can be avoided either running population-synthesis simulations with a lot more initial metallicities, which is computationally expensive, or inserting in the data-driven model a metal-licity spread, independent with redshift. This spread indeed takes into account the physical fact that DCOs evolve with different initial metallicities although they form at the same redshift.
In sections 6.1, 6.3 and 6.2, I will focus on the impact of the metallicity evolution models and population-synthesis prescriptions on MRD evolution, for each DCO type.
6.1
DNS MRD Evolution
Figure 6.1 shows the MRD across cosmic time for DNSs, evaluated with Rafelski et al. (2012) (R2012, figure 6.1(a)) and De Cia et al. (2016) (D2018, figure 6.1(b)) metallicity evolution models, for the various binary evolution prescriptions.
and α5. However, the assumed metallicity evolution model still affects the MRD. Indeed, at redshift z = 0.1 with R2012 model, the MRD evaluated with CC15α5 is lower with respect to α5. This at first sight is controversial, because lower natal kicks should increase the merger rate. I can explain this controversy by looking again at figure 5.1. I notice that CC15α5 merging efficiency is lower than α5 until Z = 0.006. R2012 metallicity evolution model predicts indeed a lower metal content at redshift zero. This trend is inverted considering D2018 model. Indeed, in figure 6.1(b), CC15α5 is higher than α5 because this metallicity evolution model predicts solar metallicity (Z = 0.02) in the local Universe.
Since η computed with α5 is independent on metallicity, consequently the cosmic MRD varies smoothly at increasing redshift, since it is driven by the SFRD evolution only. Indeed, it peaks at redshift z ∼ 2.
The MRD evaluated with the R2012 model is compatible with the LIGO-Virgo credible interval only considering the most efficient CE ejection simulated, that is α5 and low natal kicks CC15α5. D2018 metallicity model allows also CC15α3 to be consistent with LIGO-Virgo confidence interval. This means that a less efficient CE ejection can reproduce the merging rate density measured in the local universe, once considering a more robust metallicity model. I remind that De Cia model is consistent with the measured solar metallicity in the local universe, see section 4.2.2.
It is worth explaining why CC15α3 (purple line) evidently drops from
∼ 1000 mergers Gpc−3yr−1to ∼ 300 mergers Gpc−3yr−1at redshift z ∼ 5. This trend is shared by both metallicity evolution models. I can explain this fact, from figure 5.1 where is possible to notice that η evaluated with CC15α3 prescription decreases up to 8 × 10−5 at metallicity Z = 0.002.
6.2
BBH MRD Evolution
Figure 6.2(a) and 6.2(b) show a narrow spread in the BBH cosmic MRD considering different population-synthesis prescriptions. This is expected according to η which instead shows evident variations with Z (see figure 5.1). It is therefore essential the impact of metallicity on merging efficiency.
Since there have been ten BBH detections so far, the LIGO-Virgo confidence interval is narrower with respect to those of the DNSs. Reproducing the MRD in the local universe represents still a challenge. R2012 metallicity model (see figure 6.2(a)) at redshift z = 0.1 predict one order of magnitude more merger rate density with respect to the LIGO-Virgo confidence interval upper level. This is due to the fact that the metallicity at redshift zero, predicted by R2012 mode, is still too low, namely Z = 4 × 10−4 (see table 5.1).
D2018 model (figure 6.2(b)) predicts the same order of magnitude merger rate density of the LIGO-Virgo credible interval upper level at redshift zero, but still is not consistent.
From figure 6.2(a) it is possible to notice how low kicks are responsible of a mild increase of the MRD. The BH natal kicks are reduced by the fallback amount.
0 2 4 6 8 10 12 14 Lookback time [Gyr]
101 102 103 M RD D N S s (z )[ Gpc − 3y r − 1] α1 α3 α5 CC15α1 CC15α3 CC15α5 LIGO-VirgoDNSs 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) (a) DNSs - Rafelski 0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 M RD D N S s (z )[ Gpc − 3y r − 1] α1 α3 α5 CC15α1 CC15α3 CC15α5 LIGO-VirgoDNSs 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (b) DNSs - De Cia
Figure 6.1: The cosmic DNS MRD evaluated with Rafelski et al. (2012)(a) and
with respect to DNSs. This might be due to a different CE evolution for DNSs and BBHs. The data available so far cannot constrain this, which might be confirmed once the Einstein Telescope and Cosmic Explore will be operating.
6.3
BHNS MRD Evolution
The BHNS cosmic MRD (see figure 6.3(a) and 6.3(b)) is characterised by a com-bination of features peculiar of DNSs and BBHs. Indeed, it is still evident a large spread due to different population-synthesis prescriptions; and metallicity evolu-tion strongly affects the MRD. D2018 model predicts an order of magnitude less merging rate density with respect to R2012, at redshift z = 0.1. Therefore, ev-ery population-synthesis prescription coupled with D2018 is compatible with the upper-level inferred by LIGO-Virgo.
6.4
Comparisons with literature
In this section, I compare the MRD density obtained with α5 CE ejection efficiency prescription and two previous works. The purpose of these works is to predict the MRD as a function of redshift, although with different approaches.
6.4.1
Neijssel et al. (2019)
The first paper (Neijssel et al.(2019), hereafter N2019) relies on merging DCO cat-alogues obtained with COMPASS, which is another upgrade of the BSE synthesis code. In this work they do not explore the impact of various population-synthesis prescriptions on MRD. However, they implemented a metallicity evolution model characterised by a redshift-independent metallicity spread. They compared different metallicity-specific star formation rates (MSSFR) computed with the fol-lowing expression d3MSF R dtsdVcdZ = d 2MSF R dtsdVc × dP dZ(z)
whereMadau and Dickinson(2014) SFR was adopted, although N2019 left two free parameters to be constrained with gravitational-wave observations. The metallic-ity densmetallic-ity function (dPdZ(z)) was obtained by convolving a series of galaxy stellar mass functions with mass-metallicity relations, in order to take into account of the significant uncertainties whose these functions are affected. In figure 6.4 I reported their results obtained with Barrett et al. (2018) approach only. The MZ relation was given byLanger and Norman(2006) with a redshift-independent GSMF (Panter et al.,2004).
0 2 4 6 8 10 12 14 Lookback time [Gyr]
101 102 103 104 M RD B H B s (z )[ Gpc − 3y r − 1] α1 α3 α5 CC15α1 CC15α3 CC15α5 LIGO-VirgoBHBs 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z)
(a) BBHs - Rafelski et al.,2012
0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 104 M RD B H B s (z )[ Gpc − 3y r − 1] α1 α3 α5 CC15α1 CC15α3 CC15α5 LIGO-VirgoBHBs 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (b) BBHs - De Cia et al.,2018
Figure 6.2: The cosmic BBH MRD evaluated with Rafelski et al. (2012)(a) and
0 2 4 6 8 10 12 14 Lookback time [Gyr]
101 102 103 M RD B H N S (z )[ Gpc − 3y r − 1] α1 α3 α5 CC15α1 CC15α3 CC15α5 LIGO-VirgoBHNS 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z) 0.1 0.3 0.5 0.7 1.0 1.4 2 3 5 9 Cosmological Redshift (z)
(a) BHNS - Rafelski et al.,2012
0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 M RD B H N S (z )[ Gpc − 3y r − 1] α1 α3 α5 CC15α1 CC15α3 CC15α5 LIGO-VirgoBHNS 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (b) BHNS - De Cia et al.,2018
Figure 6.3: The cosmic BHNS MRD evaluated with Rafelski et al. (2012)(a) and
In figure 6.4(b), the Neijssel et al. (2019) predicts an higher BHNS merger rate density at redshift z = 0.1, but it is still under the upper limit of the LIGO-Virgo confidence interval. From redshift z > 2, the MRD tendencies are almost identical. In figure 6.4(c), the MRD obtained with R2012 metallicity model is compara-ble with Neijssel’s one, at redshift z = 0.1. In fact, N2019 state that the assumed metallicity distribution model has a similar shape to the metallicity distribution in-ferred byRafelski et al.(2012). This is a crucial fact, because it proves that despite my model and N2019 one rely on different population-synthesis simulations, MRD evolution yield similar results once the metallicity distributions are comparable. However, both models are not consistent with BBH LIGO-Virgo credible interval. D2018 metallicity model provides so far the most reliable prediction of BBH merger rate density in the local Universe.
6.4.2
Mapelli & Giacobbo (2018)
The second work I chose for this comparison isMapelli and Giacobbo (2018). They estimated the cosmic MRD of BBHs, DNSs and BHNSs by combining the results of the same population synthesis simulation presented in chapter 3, with the outputs of the Illustris-1 cosmological simulation (Vogelsberger et al., 2014). This method enables to account for the cosmic SFR density and for the metallicity evolution by means of a numerical simulation.
Figure 6.5(a) shows the DNS cosmic MRD obtained with the data-driven and numerical approaches. I notice that redshift z = 0.6 represents a turning point. Indeed, at z > 0.6 the full numerical model underestimates the merger number with respect to both data-provided cosmic evolution models. At z < 0.6 this trend is reversed.
In figure 6.5(b), the BHNS MRD is essentially the same for z > 4. At z = 0.1, the three MRD are compatible with the LIGO-Virgo upper level.
Illustris cosmological evolution predicts again a MRD which is placed between the values provided with the data-driven approach, regarding BBHs (see figure 6.5(c)). From z > 4, the MRD tendencies are again similar. D2018 provides so far the most reliable prediction of BBH cosmic MRD.
0 2 4 6 8 10 12 14 Lookback time [Gyr]
101 102 103 RDNSs (z) [G pc 3yr 1] Neijssel et al., 2019 5 R2012 5 D2018 LIGO-VirgoDNSs 0.1 0.3Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (a) DNS 0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 RBHNS (z) [G pc 3yr 1] Neijssel et al., 2019 5 R2012 5 D2018 LIGO-VirgoBHNS 0.1 0.3Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (b) BHNS 0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 RBHBs (z) [G pc 3yr 1] Neijssel et al., 2019 5 R2012 5 D2018 LIGO-VirgoBHBs 0.1 0.3Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (c) BBH
Figure 6.4: Comparison between two data-driven models that evaluate the cosmic
0 2 4 6 8 10 12 14 Lookback time [Gyr]
101 102 103 RDNSs (z) [G pc 3yr 1] Illustris 5 5 R2012 5 D2018 LIGO-VirgoDNSs 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (a) DNSs 0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 RBHNS (z) [G pc 3yr 1] Illustris 5 5 R2012 5 D2018 LIGO-VirgoBHNS 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (b) BHNSs 0 2 4 6 8 10 12 14
Lookback time [Gyr] 101 102 103 RBHBs (z) [G pc 3yr 1] Illustris 5 5 R2012 5 D2018 LIGO-VirgoBHBs 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 0.1 0.3 Cosmological Redshift (z)0.5 0.7 1.0 1.4 2 3 5 9 (c) BBHs
Figure 6.5: Comparison between the MRD across cosmic time obtained with a
Chapter 7
Conclusions
In this thesis, I presented the theoretical background for the definition of a model that places population-synthesis results into a cosmological context, in order to find the cosmic MRD. MOBSE is the binary population-synthesis code that provided catalogues of merging DCOs. It explored the impact on DCO mergers of differ-ent binary evolution prescriptions, namely three differdiffer-ent CE ejection efficiencies with high natal kicks (α1, α3 and α5) and low natal kicks (CC15α1, CC15α3 and CC15α5). The cosmological environment at different redshift was defined by data-driven functions that strongly affect the MRD.
The merging fraction of stellar-born DCOs increases, as theMadau and Dickin-son (2014) SFRD increases at increasing redshift up to a peak which can be shifted with respect to z ∼ 2, because of the delay time distributions. The MRD cosmic evolution cannot be model neglecting the SFRD. The progenitor metallicity plays also a fundamental role in determining the fate of a compact binary. Thus, I com-pared the cosmic MRD obtained with two different metallicity evolution models,
Rafelski et al.(2012) and De Cia et al. (2018).
This thesis demonstrated that DNS cosmic MRD is strongly affected by binary evolution prescriptions, while the BBH cosmic MRD is mainly determined by the av-erage metal content of the progenitor stars. The MRD is thus an essential parameter to be used to distinguish among various formation and evolution channels.
7.1
Further developments
The model I presented in my thesis, can be promptly modified to achieve new inter-esting results. The changes can be both on the population-synthesis prescriptions and on the cosmological evolution.
In figure 6.1(a) and 6.1(b), I illustrated the impact of SN kicks on DNS cosmic MRD. Recently,Giacobbo and Mapelli(2019) developed a new model for natal kicks that is in agreement with observation of galactic pulsars. They run several MOBSE population-synthesis simulations characterised by various velocity distributions of natal kicks. It is now possible to evaluate the MRD across cosmic time of DNSs with these new kick prescriptions.
One of the key open questions of GW astrophysics is determining the contribution to MRD of isolated binary evolution and dynamical environment. Di Carlo et al.
(2019) provided a catalogue of merging BBHs in young star clusters, with initial metallicity Z = 0.002. However, it is necessary to run other dynamical simulations with different initial metallicities. Indeed, it has been already assessed the impact of metallicity on cluster dynamics (Mapelli et al., 2013). Once these new data are available, in order to evaluate the MRD of dynamical evolved DCOs, is necessary to update the delay time distributions in equation 5.6, with the one obtained by dynamical simulations, leaving unchanged the SFRD and the metallicity evolution. Furthermore, I can improve my model by adding a metallicity spread, which is necessary to take into account the physical fact that galaxies with different masses have different metallicities, although born in the same redshift bin, according to the mass-metallicity relation (Tremonti et al., 2004).
Appendix A
Adopted cosmology
I assumed a standard spatially-flat 6-paramter ΛCDM cosmology. The cosmological parameters are provided by Planck 2015,Planck Collaboration et al. (2016), whose results rely on full-mission Planck observations of temperature and polarisation anisotropies of Cosmic Microwave Background (CMB). Table A.1 summarises some relevant parameters for this Thesis.
The third density parameter Ωk measures the curvature of space and is defined by the following relation
Ωm+ ΩΛ+ Ωk= 1 In this thesis, it is considered equal to zero.
Hubble Constant H0 67.8 ± 0.9 km s−1Mpc−1 Hubble Time tH0 14.4 ± 0.2 Gyr
Matter density parameter Ωm 0.308 ± 0.012 Reionization redshift zre 8.8+1.7−1.4
Table A.1: Cosmological parameters assumed in this thesis (Planck Collaboration
Appendix B
Python script
# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # DCOs m e r g i n g rate d e n s i t y as a f u n c t i o n c o s m o l o g i c a l # r e d s h i f t. C a r e f u l l y read R E A D M E.txt b e f o r e l a u n c h i n g # the code. It n e e d s i m p u t a r g u m e n t s # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # i m p o r t time s t a r t _ t i m e = time.time () i m p o r t n u m p y as np i m p o r t m a t p l o t l i b.p y p l o t as plt from a s t r o p y.c o s m o l o g y i m p o r t P l a n c k 1 5 as P l a n c k i m p o r t a s t r o p y.u n i t s as u i m p o r t os i m p o r t sys # S i m u l a t i o n s form M O B S E sim = [" A1 " , " A3 ", " A5 ", " C C 1 5 A 1 ", " C C 1 5 A 3 ", " C C 1 5 A 5 "] # I n i t i a l m e t a l l i c i t i e s Z = [" 0.0002 "," 0.0004 "," 0.0008 "," 0.0012 "," 0.0016 "," 0.002 " ," 0.004 "," 0.006 "," 0.008 "," 0.012 "," 0.016 "," 0.02 "] Z _ l i s t = [f l o a t( i ) for i in Z ] # R e d s h i f t a r r a y b i n _ z = 0.1 zMAX = 15 zMIN = -0.01# from r e d s h i f t Zmax to Zmin , i.e from f i r s t l y f o r m e d s t a r s to n o w a d a y s
# L o o k b a c k time array , used for the h i s t o g r a m of the d e l a y t i m e s
b i n _ t 1 = s o r t e d( np.a r r a y ( P l a n c k.l o o k b a c k _ t i m e ( z _ h i s t ) ) * 1 0 * * 9 ) # in yrs
b i n _ t = np.i n s e r t ( bin_t1 , 0 , 0) # b e g i n n i n g from zero # I N P U T : s e l e c t the c o m p a c t o b j e c t s of w h i c h we p e r f o r m
the c o m p u t a t i o n s ,
# it can be e i t h e r BHBs or DNSs or BHNS
obj = sys.argv [1]
# C r e a t e the file w h e r e we will save the data , in our case it will be a m a t r i x w h e r e
# the c o l u m n s will be the d i f f e r e n t set of s i m u l a t i o n s and the rows the MRD as a f u n c t i o n
# of r e d s h i f t
os.s y s t e m (" rm data / o u t p u t / MR_ "+ obj +" _ "+ m e t m o d +".txt ") #
-- -- -- >r e m o v e the p r e v i o u s file.dat
os.s y s t e m (" m k d i r data / o u t p u t ")
# D e f i n e the m a t r i x w h e r e we save our data
if obj = = ’ DNSs ’: Z _ a v e r a g e = 0.002 else: Z _ a v e r a g e = 0.0002 elif m e t m o d = = ’ H i g h e s t M e t ’: Z _ a v e r a g e = 0.02 elif m e t m o d = = ’ D e C i a ’: Z _ a v e r a g e =0.0 2 * 1 0 * * (-0.18-0.24* z ) elif m e t m o d = = ’ D e C i a 2 ’: Z _ a v e r a g e =0.0 2 * 1 0 * * (-0.24* z ) r e t u r n Z _ a v e r a g e
# Loop over the s i m u l a t i o n s
for s in r a n g e(len( sim ) ) :
p r i n t( sim [ s ])
# Load the t a b l e with the m e r g e r per unit mass
[ Zsim , rb , rn , rbn ] = np.l o a d t x t (’ data / ’+ sim [ s ] + ’ / t a b l e _ m e r g e r _ p e r _ u n i t _ m a s s.txt ’, s k i p r o w s=1 , u n p a c k=True ) # D e c i d e what o b j e c t s if( obj = = ’ BHBs ’) : R _ c o r = rb objN = 1 elif( obj = = ’ DNSs ’) : R _ c o r = rn objN = 0 elif( obj = = ’ BHNS ’) : R _ c o r = rbn objN = 2 w = (len( z _ h i s t ) , len( z _ h i s t ) ) v = np.z e r o s ( w ) # it is a s q u a r e m a t r i x made up by zeros , it is used for
# s a v i n g the h i s t o g r a m v a l u e s i = 0 # this i n d e x runs on f o r m a t i o n r e d s h i f t NM = np.z e r o s (len( z _ h i s t ) ) # = SFR * dt , at g i v e n f o r m a t i o n r e d s h i f t i n d e x = np.z e r o s (len( z _ h i s t ) , d t y p e = int) # i n d i c a t e s the m e t a l l i c i t y a r r a y i n d e x
# Loop over the f o r m a t i o n r e d s h i f t for zf in z _ h i s t :
# Star f o r m a t i o n rate at a g i v e n r e d s h i f t
SFR = 0.0 1 5 * ( 1 + zf-b i n _ z ) **(2.7) / ( 1 + ( ( 1 + zf
# Msun yr-1 Mpc-3 m i n _ d t = P l a n c k.l o o k b a c k _ t i m e ( zf ) / u.Gyr * 1 0 * * 9 # In yr m a x _ d t = P l a n c k.l o o k b a c k _ t i m e ( zf-b i n _ z ) / u.Gyr * 1 0 * * 9 # In yr dt = m i n _ d t - m a x _ d t
# N u m b e r of s o l a r m a s s e s per unit v o l u m e [ at the end will be c o n v e r t e d in Gpc ^{-3}] NM [ i ] = dt * SFR # Msun Mpc-3 Z _ a v e r a g e = metz ( zf ) # find the c l o s e s t Z in my s i m u l a t i o n s err = 100. for met in Z _ l i s t :
if err > np.abs( Z _ a v e r a g e - met ) : err = np.abs( Z _ a v e r a g e - met ) i n d e x [ i ] = Z _ l i s t.i n d e x ( met )
# Load the data a b o u t the d e l a y time , once we c h o s e the
# g i v e n m e t a l l i c i t y at a g i v e n f o r m a t i o n r e d s h i f t
[ ID , m1in , m2in , mrem1 , mrem2 , mrem , td , sep , ec ] = np.
l o a d t x t (’ data / ’+ sim [ s ] +’ / d a t a _ ’
+ obj +’ _ ’+ Z [ i n d e x [ i ]]+’.txt ’, s k i p r o w s=1 , u n p a c k=
True )
# C o m p u t e the h i s t o g r a m of the m e r g e r time
t _ f o r m = np.a r r a y ( P l a n c k.l o o k b a c k _ t i m e ( zf ) / u.Gyr * 1 0 * * 9 ) # yrs
t _ m e r g = t _ f o r m - td * 1 0 * * 9 # yrs
# t _ m e r g must be p o s i t i v e! we fix the b i n n i n g
d t S o r t e d = s o r t e d( t_merg , key=f l o a t)
# e v a l u a t e the h i s t o g r a m
[ vv , base ] = np.h i s t o g r a m ( dtSorted , bins = b i n _ t )
# I e v a l u a t e the m e r g i n g rate as a f u n c t i o n of r e d s h i f t
rate = 0
for k in r a n g e(0 ,len( z _ h i s t ) ,1) :# k s e l e c t s the
m e r g i n g r e d s h i f t rate = 0 p r i n t(’ z m e r g i n g = ’, z _ h i s t [ k ]) for q in r a n g e(0 , k + 1 , 1) : # k +1 in o r d e r to have d i a g o n a l v a l u e s rate = rate + v [ k , q ]* NM [ q ]* R _ c o r [ i n d e x [ q ]] # v [ k , q ] is the f r a c t i o n of m e r g i n g c o m p a c t o b j e c t f o r m e d at r e d s h i f t q # and m e r g i n g at r e d s h i f t k. q is a l w a y s s m a l l e r or e q u a l than k . k = 0 # c o r r e s p o n d s to r e d s h i f t z = Zmax , i.e. 15. We need the n u m b e r of c o m p a c t
# o b j e c t per unit mass , that ’ s the why we need to m u l t p l y it by R _ c o r w h i c h
# is the t o t a l n u m b e r of m e r g i n g c o m p a c t o b j e c t s per unit mass
MR_z [ s , k ] = rate /( b i n _ t [len( z _ h i s t )-k ]-b i n _ t [len ( z _ h i s t )-k-1]) * 1 0 * * 9 # Gpc ^-3 yr ^-1
# Save MR_z info in a file and make a plot
np.s a v e t x t (" MR_ "+ obj +" _ "+ m e t m o d +".txt ", np.t r a n s p o s e ( np.
a s m a t r i x ( MR_z ) ) )
os.s y s t e m (" mv MR_ "+ obj +" _ "+ m e t m o d +".txt data / o u t p u t ") e l a p s e d _ t i m e = time.time ()-s t a r t _ t i m e
p r i n t(’ The code took ’, e l a p s e d _ t i m e /60 , ’ m i n u t e s to run
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